Paper VI
Benchmarking two-photon absorption
cross sections: Performance of
CC2 and CAM-B3LYP
M. T. P. Beerepoot, D. H. Friese, N. H. List, J. Kongsted and K. Ruud
Phys. Chem. Chem. Phys. 17 (2015), 19306–19314.
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Benchmarking two-photon absorption cross
sections: performance of CC2 and CAM-B3LYP
Maarten T. P. Beerepoot,*a Daniel H. Friese,a Nanna H. List,b Jacob Kongstedb and
Kenneth Ruuda
We investigate the performance of CC2 and TDDFT/CAM-B3LYP for the calculation of two-photon
absorption (TPA) strengths and cross sections and contrast our results to a recent coupled cluster
equation-of-motion (EOM-EE-CCSD) benchmark study [K. D. Nanda and A. I. Krylov, J. Chem. Phys., 2015,
142, 064118]. In particular, we investigate whether CC2 TPA strengths are significantly overestimated
compared to higher-level coupled-cluster calculations for fluorescent protein chromophores. Our
conclusion is that CC2 TPA strengths are only slightly overestimated compared to the reference EOM-EE-
Received 4th June 2015,
Accepted 25th June 2015
CCSD results and that previously published overestimated cross sections are a result of inconsistencies in
DOI: 10.1039/c5cp03241e
other hand, are found to be 1.5 to 3 times smaller than the coupled-cluster reference for the molecular
systems considered. The unsatisfactory performance of TDDFT/CAM-B3LYP can be linked to an
www.rsc.org/pccp
underestimation of excited-state dipole moments predicted by TDDFT/CAM-B3LYP.
the conversion of the TPA strengths to macroscopic units. TDDFT/CAM-B3LYP TPA strengths, on the
1 Introduction
Two-photon absorption (TPA)—that is, a simultaneous absorption
of two photons1—is a non-linear optical process with an intensity
depending on the square of the incoming light. Searching for
molecules with high TPA cross sections is an active field of
research2,3 and the interplay between experiment and theory is
paramount for a good understanding of structure–property
relations.2,4
Quantum chemical calculation of TPA strengths has been
implemented at various levels of theory in different software
packages and has recently gained significant interest. The
implementation in the Dalton software package5 was described
by Hättig, Christiansen and Jørgensen6 for coupled-cluster
methods and by Sałek et al.7 for time-dependent density functional
theory (TDDFT). More recently, Friese, Hättig and Ruud8 implemented TPA based on the resolution-of-identity CC2 (RI-CC2) in
a development version of Turbomole,9 thereby enabling the
calculation of TPA at the coupled-cluster level for medium-sized
molecules. Further, calculation of TPA strengths was implemented
in QChem10,11 for the algebraic diagrammatic construction method
by Knippenberg et al.12 and very recently for equation-of-motion for
excitation energies CCSD (EOM-EE-CCSD) by Nanda and Krylov.13
a
Centre for Theoretical and Computational Chemistry, Department of Chemistry,
University of Tromsø—The Arctic University of Norway, N-9037 Tromsø, Norway.
E-mail: [email protected]; Tel: +47 77623103
b
Department of Physics, Chemistry and Pharmacy, University of Southern Denmark,
DK-5230 Odense M, Denmark
19306 | Phys. Chem. Chem. Phys., 2015, 17, 19306--19314
In addition, Salem and Brown14 have reported TDDFT TPA cross
sections calculated in GAMESS.15 All these different implementations span the spectrum from computationally efficient to very
accurate, with TDDFT allowing for efficient calculations on large
molecules and the coupled-cluster hierarchy of methods allowing
for systematic improvement of the accuracy. CC2 and TDDFT have
so far been benchmarked against higher-order coupled-cluster
methods only for very small organic molecules of up to six atoms.16
Benchmarking CC2 and TDDFT TPA strengths against higher-level
methods is important in order to assess the accuracy of these
methods in work on larger molecules, for which more accurate
calculations are not yet feasible. The largest molecule studied using
EOM-EE-CCSD by Nanda and Krylov is 4-(p-hydroxybenzylidene)1,2-dimethylimidazolin-5-one (HBDI) with 28 atoms.13 CC2 TPA
cross sections have been reported for molecular systems of more
than twice that size: a truncated molecular tweezer complex consisting of 78 atoms8 and the YFP–Tyr203 dimer with 62 atoms.17
TDDFT TPA cross sections have been reported for molecular
systems with more than 100 atoms, such as the fullerene–buckycatcher complex in ref. 18.
Very recently, Nanda and Krylov reported TPA strengths for
medium-sized (20 to 28 atoms) neutral fluorescent protein
chromophores using the EOM-EE-CCSD method.13 By comparing
their calculated cross sections to published CC2 results using the
same geometry of the HBI chromophore,14 they noted a strong
overestimation (a factor of 4.4) by CC2.13 The suggested explanations for this discrepancy were a strong overestimation of TPA
strengths for CC2 or a mistake in the conversion to macroscopic
units.13 In addition, a previous study by three of us on CC2 and
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TDDFT/CAM-B3LYP cross sections on intermolecular chargetransfer excitation in the yellow fluorescent protein also reported
CC2 TPA cross sections to be larger than those calculated with
TDDFT/CAM-B3LYP.17
The aim of this study is to assess the quality of CC2 and
TDDFT/CAM-B3LYP TPA strengths by comparing with the
recent benchmark study by Nanda and Krylov13 and to inspect
whether CC2 calculations do indeed strongly overestimate the
TPA strength. Comparison between different methods are often
made on the basis of macroscopic rather than microscopic
units, leading to extra challenges and complications for benchmarking work. Therefore we begin in Section 2 by addressing
different factors that need to be considered in order to make a
correct comparison between TPA cross sections in macroscopic
units. Following this, we compare the TPA strengths of a set of
fluorescent protein chromophores computed with CC2 and
TDDFT/CAM-B3LYP to those obtained with EOM-EE-CCSD13
and previously reported CC214 cross sections, using the same
molecular geometries as in the reference works.
2 Conversion to macroscopic units
The macroscopic TPA cross section sTPA in cgs units can be
obtained from the rotationally averaged TPA strength hdTPAi in
atomic units using19
sTPA ¼
Np3 aa0 5 o2 TPA d
gð2o; o0 ; GÞ;
c
(1)
where N is an integer value (see Section 2.3), a is the fine
structure constant, a0 the Bohr radius, o the photon energy in
atomic units, c the speed of light and g(2o, o0, G) the lineshape
function describing spectral broadening effects (see Section 2.2).
The common unit for TPA cross sections is GM (originally
introduced as ‘maria’)20 in honour of the work of Maria GöppertMayer,1 with 1 GM corresponding to 1050 cm4 s photon1. When
deriving eqn (1) without taking into account factors related to
the comparison to a specific experimental setup, one obtains
N = 8.19,21 As will be elaborated in Section 2.3, this corresponds
to a so-called double-beam experiment. In passing, we note that
the original work of Peticolas, describing the conversion of
multiphoton absorption strengths to macroscopic units, contains a typo in the equation for the transition strength for twophoton processes (eqn (10) in that work).19 In fact, the exponent
2 of the term 2ph/V is omitted in eqn (10), which could make
one believe that eqn (1) depends on 4p2 rather than 8p3 for a
double-beam experiment. It is clear that this is a typo by
comparing that eqn (10) to eqn (17) on two-photon absorption
and eqn (28) on three-photon absorption in the same work.19
This indicates that a factor (2p)3 should appear in the final
conversion for two-photon absorption and a factor (2p)4 for
three-photon absorption. This factor (2p) j+1 for j-photon
absorption is also derived in ref. 21.
One should be aware that o in eqn (1) is the photon energy,
which in the degenerate TPA case corresponds to half the excitation
energy. Defining o as the excitation energy (2o) rather than the
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photon energy (as is done in ref. 14 and 22) results in TPA cross
sections higher by a factor of 4.
In the following we will describe the prefactor from the
rotational averaging of the TPA strength (Section 2.1), the
lineshape function g(2o, o0, G) (Section 2.2), and the type of
experiment that is compared to (Section 2.3) in more detail with
particular emphasis on what one needs to be aware of when
comparing TPA strengths in macroscopic units.
2.1
The prefactor of the rotational averaging
The rotationally averaged TPA strength hdTPAi in atomic units
can be obtained from the TPA transition moment S as23–25
TPA 1 X
¼
d
ðFSaa Sbb þ GSab Sab þ HSab Sba Þ;
30 ab
(2)
where the sum is over the Cartesian components a and b and
with the bar indicating complex conjugation. In the electric
dipole approximation, the (a, b)’th component of the TPA
transition moment Sif between the initial state i and final state
f is defined as26
b j f i hijmb jnihnjm
a j f i
1 X hijma jnihnjm
if
; (3)
Sab
ðo 1 ; o 2 Þ ¼
þ
h nai
oni o1
oni o2
where hi|ma|ni is the transition dipole moment between the
electronic states i and n along the Cartesian axis a, oni the
associated excitation energy and o1 and o2 the frequencies of
photon 1 and 2. ma = ma h0|ma|0i, i.e.hn|
ma| f i is the difference
dipole moment between the ground and excited state if n = f 27
and a transition dipole moment if n a f.
For linearly polarized light with parallel polarization (F = G =
H = 2)23 and two photons of the same energy, eqn (2) reduces to
TPA 1 X
d
¼
ð2Sab Sab þ Saa Sbb Þ:
15 ab
Alternatively, hdTPAi can be defined as28
TPA X
d
ðFSaa Sbb þ GSab Sab þ HSab Sba Þ;
¼
(4)
(5)
ab
or as14,22
X
dTPA ¼
ð2Sab Sab þ Saa Sbb Þ;
(6)
ab
in which case the prefactor of the rotational averaging is
included in the conversion factor as14,22,28,29
sTPA ¼
4p3 aa0 5 o2 TPA gð2o; o0 ; GÞ:
d
15c
(7)
One should however be aware that the same eqn (7) results
from the use of N = 4 in ref. 14 and 22 (where hdTPAi is defined
as in eqn (6), leaving out a factor of 1/15) and from the use of
N = 8 in ref. 28 (where hdTPAi is defined as in eqn (5), leaving out
a factor of 1/30). Indeed, it is unclear whether N = 4 or N = 8 is
used in ref. 29 because the conversion is defined as in eqn (7)
without a definition of hdTPAi. This points to the importance of
publishing not only the equation used for the conversion factor
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(e.g. eqn (1)), but also the way in which the rotational average is
defined (e.g. eqn (4)).
2.2
The lineshape function
The lineshape function g(2o, o0, G) with broadening factor G
describes broadening effects, correcting for the infinitely sharp
calculated vertical excitations and allowing for comparison to
experimental peaks, in which rovibrational excitations and
collisional dynamics also play a role.26 This is usually done in
a phenomenological way by introducing an empirical damping
parameter G to describe the spectral broadening of an excitation. In many gas-phase calculations, a Lorentzian lineshape
function is used for g(2o, o0, G),
Lð2oÞ ¼
1
G
;
p ð2o o0 Þ2 þðGÞ2
(8)
with o the photon energy, o0 the excitation energy and G the
half width at half maximum (HWHM). The function is normalized to one. By setting o = o0/2, the maximum of the Lorentzian
is obtained as
Lðo0 Þ ¼
1
:
pG
(9)
When this maximum is inserted for g(2o, o0, G) in eqn (1), one
obtains
sTPA ¼
Np2 aa0 5 o2 TPA d
cG
(10)
for the TPA cross section at the calculated excitation energy
with hdTPAi defined as in eqn (2) or (4). We note that the
exponent of p is 3 in eqn (1) and 2 in eqn (10), which is
sometimes overseen when the equation is printed.22
In solution, a Gaussian lineshape function is most commonly
used for g(2o, o0, G),30
"
pffiffiffiffiffiffiffiffi
#
ln 2
2o o0 2
p
ffiffiffi
Gð2oÞ ¼
;
(11)
exp ln 2
G
G p
again normalized to one and with o the photon energy, o0 the
excitation energy and G the HWHM. The maximum of the
Gaussian is obtained at o = o0/2 as
pffiffiffiffiffiffiffiffi
ln 2
(12)
Gðo0 Þ ¼ pffiffiffi :
G p
When this maximum is inserted for g(2o, o0, G) in eqn (1), one
obtains
pffiffiffiffiffiffiffiffi
5
ln 2 Np2 aa0 5 o2 TPA :
(13)
sTPA ¼
d
cG
Thus, when using the same broadening factor G, the maxima of the
Lorentzian and Gaussian broadening functions have a different
value. The Lorentzian function has a broader base and the Gaussian
pffiffiffiffiffiffiffiffiffiffiffi
function has a higher maximum by a factor of p ln 2 1:48,
see Fig. 1a.
One should be aware that G can also be defined as the full
width at half maximum (FWHM) instead of the HWHM (FWHM =
2HWHM). The use of G = 0.2 eV as the FWHM is equivalent to a
19308 | Phys. Chem. Chem. Phys., 2015, 17, 19306--19314
Fig. 1 (a) Lorentzian (eqn (8)) and Gaussian (eqn (11)) broadening functions with HWHM = 0.1, (b) Lorentzian broadening functions with G = 0.2
as FWHM and as HWHM and (c) Lorentzian broadening functions with
different values of G as HWHM.
HWHM of 0.1 eV when one substitutes G for G/231–34 in eqn (8)
or (11). If one uses 0.1 eV as FWHM,35 the resulting TPA cross
section sTPA is twice as high compared to work using 0.1 eV as
HWHM, see Fig. 1b and eqn (9) and (12). We also point out that
the use of eqn (10) with N = 818,36–38 can in principle result from
either the use of eqn (1) with N = 8 and G defined as the HWHM
or with N = 4 and G defined as the FWHM.
The broadening effects are different for each excited state,26
which can also be taken into account in theoretical work.30,32,36,39
What is usually done, however, is choosing a single empirical
parameter for G, often chosen to be 0.1 eV,13,14,18,21,22,28,29,37,38
but also other broadening factors have been used, usually taken
from the broadening of a specific peak in an experimental
spectrum.2,31,34,35 We note that an alternative approach used
for the calculation of TPA strengths is given by damped response
theory,40 in which an imaginary factor ig is added to the optical
frequency o to incorporate broadening effects. In work on damped
response theory with the complex polarization propagator such as
ref. 40, g is used for the HWHM and G for the FWHM.
One should be cautious when comparing TPA cross sections
computed with different broadening factors as they affect also
the maximum of the peak, see Fig. 1c. The discussion here
makes it sufficiently clear that giving the value of G and
specifying the type of broadening and whether G is HWHM
or FWHM is important for the reproducibility of published TPA
cross sections.
2.3
Comparison with experiment
The use of different integers N in eqn (1) is among other
reasons due to comparison to different experimental setups.
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Schematic overview of the value for N needed in eqn (1).
When deriving eqn (1), one obtains N = 8.19,21 This corresponds to
an experimental setup with two laser sources (a so-called doublebeam experiment41), thus in principle allowing for two photons
with different energies. The use of N = 8 in eqn (1), however, is not
correct when comparing calculated cross sections to the photon
dissipation rate in single-beam TPA experiments.30–33,39,42,43 In fact,
the photon dissipation rate is in this case twice the calculated twophoton absorption rate because two photons together excite the
molecule. Thus, one needs N = 16 in eqn (1) to compare to singlebeam experiments.31,33,39,42,43 When two lasers are used as the
photon sources—as intended in e.g. ref. 19 and 21—the photon
dissipation rate of each of the lasers equals the two-photon
absorption rate in the sample.
Furthermore, the TPA transition moment (see eqn (3)) can
be defined in different ways.31–33,39,41–43 The correct definition
of the two-photon transition moment Sif to compare to singlebeam experiments (such as the z-scan technique3) is half the
one in eqn (3),31–33,39,41–43
if
Sab
ðoÞ ¼
b j f i
1 X hijma jnihnjm
;
h nai
oni o
(14)
where there is only one photon with frequency o. The use of
eqn (14) instead of eqn (3) leads to hdTPAi—which depends on
terms of S times S% —lower by a factor of 4. To correct for this
factor of 4 in hdTPAi, a factor of N = 4 (instead of N = 16) in the
conversion to macroscopic units has been used in combination
with hdTPAi calculated using S in eqn (3) to obtain a sTPA that
can be compared to single-beam experiments.30,32,34 This is the
strategy implemented in e.g. the Dalton program,5,44 where S is
defined as in eqn (3). The choice of the right value of N in
eqn (1) is summarized in Fig. 2.
3 Computational details
We present CC2 and TDDFT/CAM-B3LYP one- and two-photon
absorption calculations for the four neutral fluorescent protein
chromophores in Fig. 3. The HBI (4-( p-hydroxybenzylidene)imidazolidin-5-one) and HBDI (4-( p-hydroxybenzylidene)-1,2dimethylimidazolin-5-one) molecules are models for the neutral
green fluorescent protein chromophore. The chromophore of the
cyan fluorescent protein (CFP) is similar to HBI with an indole ring
in place of the phenol ring. The chromophore of the phosphorescent
yellow protein (PYP) is p-coumaric acid. The chromophores
of fluorescent proteins are frequently studied using TPA both
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Fig. 3
Molecular structures of the chromophores used in this study.
experimentally45,46 and computationally13,14,17,22,37,38,47,48 due
to their relevance in biological imaging.
All calculations are gas-phase calculations on vertical excitations.
Calculations with the approximate coupled-cluster singles and
doubles model (CC2)49 were done in Turbomole9 using the
resolution-of-identity (RI) approximation.8,50,51 The RI approximation has been shown to give insignificant errors compared
to the basis-set error for excitation energies50 and one-photon
oscillator strengths.51 1s-orbitals of non-hydrogen atoms were
kept frozen in the correlation treatment. TDDFT calculations
were done using Dalton5,7,44 with the CAM-B3LYP exchange–
correlation functional52 with a = 0.19, b = 0.46 and m = 0.33. The
range-separated functional CAM-B3LYP has an average error that is
independent of the degree of charge transfer in the excitations
investigated.29,47,53 Furthermore, it yields excitation energies47 and
oscillator strengths,54 also considered in this work, in very good
agreement with coupled-cluster results for neutral chromophores.
It should be noted that the parameters of the functional are
optimized using a test set of only neutral chromophores,52 making
the transferability to charged systems unknown.
Basis sets from the Pople (6-31+G*)55–57 and Dunning (aug-ccpVDZ, d-aug-cc-pVDZ and aug-cc-pVTZ)58 families were used to
ensure reliable comparison to previous works.13,14 The auxiliary
basis sets aug-cc-pVDZ (for molecular basis sets 6-31+G*, aug-ccpVDZ and d-aug-cc-pVDZ) and aug-cc-pVTZ (for molecular basis
set aug-cc-pVTZ) were used for all RI-CC2 calculations.59 To
compare to the work of Nanda and Krylov,13 we removed the
d-functions from the second set of augmentation functions from
the d-aug-cc-pVDZ basis set. We will refer to this modified basis
set as mod-d-aug-cc-pVDZ in Section 4.
Molecular structures were taken from reference works to ensure
that differences in calculated one-photon and two-photon absorption properties result only from different methods and not from
using different molecular geometries. We will therefore only compare our results to other work where exactly the same molecular
geometries were used13,14 and not to works where different geometries of the same molecules were studied.22,37,47,48 The geometries of
HBI and CFP were taken from Salem and Brown,14 where the
geometries were optimized with DFT using the PBE0 exchange–
correlation functional and the 6-31+G** basis set. The geometries of
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HBDI and PYP were taken from Nanda and Krylov,13 where the
geometries were optimized with RI-MP2 and the aug-cc-pVDZ and
aug-cc-pVTZ basis sets, respectively. All chromophores have Cs symmetry, which was also exploited in the CC2 calculations. We limit our
study to the lowest excitation of the chromophores (lowest two
excitations for PYP), which is of A0 symmetry and p - p* character.
The excitation to the 2A0 state of PYP and the considered excitations in
HBI, HBDI and CFP are dominated by the HOMO–LUMO transition
with both orbitals involved delocalized over the whole conjugated
system of the chromophore. The 1A0 excitation of PYP has contributions from the HOMO and LUMO as well as from an occupied p and a
virtual p* orbital that are both located on the phenol ring.
Rotational averaging of the TPA transition moments was
done using eqn (4) with Sab defined as in eqn (3). Conversion to
GM units was done using eqn (1) with N = 4 and a Lorentzian
lineshape function with a HWHM of 0.1 eV was used to ensure
comparability to reference work.13 Since we only report cross
sections at resonant conditions (o = o0/2), the conversion
corresponds to eqn (10) with N = 4 and G = 0.1 eV.
4 Results
Calculated excitation energies and one- and two-photon absorption properties for the four molecules in Fig. 3 are tabulated in
Tables 1–4 (HBI, HBDI, CFP and PYP) together with reference
values from the literature.13,14
The calculated CC2/6-31+G* TPA cross sections sTPA for HBI
(Table 1) and CFP (Table 3) are a factor of 4 lower than the CC2 cross
section in ref. 14 using the same molecular geometry. This shows
that the conversion factor used here and by Nanda and Krylov13 is
different from the one used by Salem and Brown14 by a factor of 4.
The overestimation of the TPA cross section of the first A0 state of the
HBI chromophore by CC2/6-31+G* in comparison to EOM-EE-CCSD/
6-31+G* is thus not a factor of 4.4,13 but a factor of 1.1. We find that
CC2 also overestimates EOM-EE-CCSD TPA strengths hdTPAi with the
mod-d-aug-cc-pVDZ basis set by a factor of 1.16 for HBDI and by
factors of 1.25 and 1.34 for the 2A0 and 1A0 states in PYP, respectively
(see Tables 2 and 4). Differences in excitation energies, oscillator
strengths and TPA cross sections between the aug-cc-pVDZ and
mod-d-aug-cc-pVDZ basis sets calculated with CC2 are negligible (see
Tables 1, 2 and 4).
Table 1 Excitation energy 2o and DE, oscillator strength f, TPA strength
hdTPAi and TPA cross section sTPA for the 1A 0 state of the HBI chromophore. All calculations are performed using the same molecular geometry
taken from ref. 14
Method
Basis set
EOM-EE-CCSD
CC2
CC2
CAM-B3LYP
CC2
CAM-B3LYP
EOM-EE-CCSD
CC2
CC2
CAM-B3LYP
mod-d-aug-cc-pVDZ
mod-d-aug-cc-pVDZ
aug-cc-pVDZ
aug-cc-pVDZ
aug-cc-pVTZ
aug-cc-pVTZ
6-31+G*
6-31+G*
6-31+G*
6-31+G*
2o
[a.u.]
DE
[eV] f
hdTPAi sTPA
[a.u.] [GM] Ref.
14.68 13
3345 16.75
3358 16.83
994 4.90
3527 17.60
983 4.85
18.3 13
3.80
80
14
0.1399 3.81 0.85 3790 20.11
0.1368 3.72 0.75 1057 5.37
0.1359
0.1359
0.1348
0.1356
0.1348
3.70
3.70
3.67
3.69
3.67
0.80
0.80
0.72
0.81
0.72
19310 | Phys. Chem. Chem. Phys., 2015, 17, 19306--19314
Table 2 Excitation energy 2o and DE, oscillator strength f, TPA strength
hdTPAi and TPA cross section sTPA for the 1A 0 state of the HBDI chromophore. All calculations are performed using the same molecular geometry
taken from ref. 13
Method
Basis set
2o
[a.u.]
DE
[eV] f
hdTPAi sTPA
[a.u.] [GM] Ref.
EOM-EE-CCSD
CC2
CC2
CAM-B3LYP
CC2
CAM-B3LYP
CAM-B3LYP
mod-d-aug-cc-pVDZ
mod-d-aug-cc-pVDZ
aug-cc-pVDZ
aug-cc-pVDZ
aug-cc-pVTZ
aug-cc-pVTZ
6-31+G*
0.1355
0.1356
0.1354
0.1353
0.1354
0.1372
3.97
3.69
3.69
3.68
3.68
3.68
3.73
978
1135
1136
490
1281
491
516
0.77
0.75
0.75
0.74
0.76
0.74
0.77
5.64 13
5.66
5.67
2.44
6.37
2.44
2.64
Table 3 Excitation energy 2o and DE, oscillator strength f, TPA strength
hdTPAi and TPA cross section sTPA for the 1A 0 state of the CFP chromophore. All calculations are performed using the same molecular geometry
taken from ref. 14
DE
[eV]
hdTPAi sTPA
[a.u.] [GM]
Method
Basis set
2o
[a.u.]
CAM-B3LYP
CC2
CAM-B3LYP
CC2
CC2
CAM-B3LYP
aug-cc-pVTZ
aug-cc-pVDZ
aug-cc-pVDZ
6-31+G*
6-31+G*
6-31+G*
0.1287 3.50 0.61 1659
0.1284 3.49 0.69 5197
0.1287 3.50 0.61 1672
3.58
0.1318 3.59 0.73 5411
0.1305 3.55 0.64 1674
f
Ref.
7.46
23.24
7.51
102
14
25.50
7.73
TDDFT/CAM-B3LYP TPA strengths hdTPAi, on the contrary, are
much lower than the coupled-cluster results. The difference depends
on the molecule and is for different basis sets a factor of 3.4 to 3.6 for
HBI, 2.3 to 2.6 for HBDI, 3.1 to 3.2 for CFP and 1.9 for PYP compared
to CC2 calculations. The underestimation of CAM-B3LYP/aug-ccpVDZ in comparison to EOM-EE-CCSD/d-aug-cc-pVDZ is a factor of
2.00 for HBDI and 1.50 for PYP for hdTPAi and a factor of 3.00 for HBI
for sTPA (for which no hdTPAi is given in ref. 13). The TDDFT/CAMB3LYP cross sections are more or less converged when using the
aug-cc-pVDZ basis set, as was also shown in a previous study of oneand two-photon properties on the neutral and anionic structures of
the GFP chromophore using the crystal structure geometry.37 TPA
cross sections are slightly larger with the 6-31+G* basis set, which
was previously found to be the case for the neutral but not for the
anionic GFP chromophore.37
In contrast to the TPA strengths, the CAM-B3LYP oscillator
strengths are in good agreement with the coupled-cluster results with
very similar values for HBDI, a slight underestimation for HBI and
CFP and a modest overestimation for PYP, none of which exceeds
20%. This is in agreement with a more thorough analysis of Caricato
et al., who found that CAM-B3LYP provides the best agreement with
EOM-CCSD oscillator strengths among the set of density functionals
investigated for 11 small organic neutral molecules.54
5 Discussion
The most important observation from the numerical results
presented in the previous section is that CC2 TPA strengths
agree well with the benchmark EOM-EE-CCSD results, whereas
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Table 4 Excitation energy 2o and DE, oscillator strength f, TPA strength hdTPAi and TPA cross section sTPA for the 1A 0 and 2A 0 states (1A 0 state for
CAM-B3LYP) of p-coumaric acid (PYP chromophore). The states are labeled to correspond to the order of the EOM-EE-CCSD states in ref. 13 and are
therefore not in order of ascending energy for the CC2 calculations. All calculations are performed using the same molecular geometry taken from ref. 13
Method
Basis set
EOM-EE-CCSD
mod-d-aug-cc-pVDZ
CC2
mod-d-aug-cc-pVDZ
CC2
aug-cc-pVDZ
CAM-B3LYP
CC2
aug-cc-pVDZ
aug-cc-pVTZ
CAM-B3LYP
CAM-B3LYP
aug-cc-pVTZ
6-31+G*
State
2o [a.u.]
DE [eV]
f [a.u.]
hdTPAi [GM]
sTPA
Ref.
4.48
4.75
4.37
4.63
4.37
4.63
4.31
4.36
4.61
4.30
4.36
0.22
0.53
0.52
0.27
0.52
0.28
0.65
0.52
0.27
0.65
0.68
1224
2631
3281
1646
3278
1645
1758
3239
1596
1706
1953
9.02
21.47
22.91
12.94
22.91
12.93
11.94
22.51
12.40
11.57
13.57
13
13
0.1604
0.1702
0.1605
0.1703
0.1582
0.1601
0.1692
0.1581
0.1601
0
1A
2A 0
2A 0
1A 0
2A 0
1A 0
2A 0
2A 0
1A 0
2A 0
2A 0
the TDDFT/CAM-B3LYP TPA strengths are significantly
underestimated.
In a thorough comparison of TPA strengths of small molecules of up to six atoms by Paterson et al., CC2 was compared to
the higher-order methods CCSD and CC3.16 For some of the
investigated transitions, CC2 was observed to overestimate
CCSD and CC3 TPA strengths, but not by more than a factor
of 2. CCSD, on the other hand, was shown on average to
overestimate TPA strengths slightly in comparison to CC3.16
Combined with the results presented in this work, this seems to
indicate that CC2 performs generally well in the calculation of
TPA strengths with the possible exception of specific transitions that require an accurate description of double excitations.
It is interesting to understand whether the underestimation
of TDDFT/CAM-B3LYP TPA strengths is specific to the limited
number of excitations in the (neutral) molecules investigated
here, or more general. While the number of studies reporting
TPA cross sections calculated with TDDFT is high, few of them
address the accuracy compared to more accurate methods. This
is in part related to the lack of efficient publicly available codes
to compute coupled-cluster TPA strengths. At least four previous studies have included a comparison of CC2 and TDDFT/
CAM-B3LYP TPA strengths. Paterson et al.16 included different
density functionals in their comparison to the coupled-cluster
hierarchy of methods on small molecules of up to six atoms.
They found that CAM-B3LYP generally outperforms LDA, BLYP
and B3LYP but overestimates most of the TPA strengths.16
Friese, Ruud and Hättig discuss the agreement between CC2
and TDDFT/CAM-B3LYP mainly in a qualitative way, but report
cross sections of CAM-B3LYP/cc-pVDZ both higher and lower
than CC2/cc-pVDZ for qualitatively similar electronic transitions.8
In a previous work, three of us have inspected a p-stacking system
of an extended anionic HBDI chromophore and a tyrosine residue
found in the yellow fluorescent protein.17 CC2/aug-cc-pVDZ TPA
cross sections were found to be much higher than CAM-B3LYP/
aug-cc-pVDZ cross sections. In particular, the TPA cross section of
the p - p* excitation located on the chromophore was lower by a
factor of 6.3 for the CAM-B3LYP calculations. Comparing this with
the factor of 2.3 in this work (Table 2), we note a significant
difference that is presumably due to using the neutral or anionic
HBDI chromophore. Salem and Brown14 report CC2/6-31+G* TPA
This journal is © the Owner Societies 2015
cross sections for the same p - p* transition of five neutral
chromophores, including HBI and CFP. Their CAM-B3LYP/
6-31+G** gas-phase TPA cross sections (in their ESI) are smaller
by a factor of 1.3 for anionic HBI, 3.6 for neutral HBI, 3.3 for CFP
and 2.5 and 4.5 for two different protonation states of the chromophore of the blue fluorescent protein. Similar or larger underestimations are reported for the B3LYP and PBE0 exchange–
correlation functionals.
The underestimation of CAM-B3LYP TPA strengths can be
understood by its underestimation of excited-state dipole
moments.60–62 Eriksen et al. found that the ground-state dipole
moment of para-nitroaniline is overestimated by CAM-B3LYP,
while the excited-state dipole moment of the charge-transfer
state is dramatically underestimated, resulting in a much too
small difference dipole moment.60 Zaleśny and co-workers
found that the problems with excited-state dipole moments
are not limited to a specific molecule or a particular functional,
but constitute a general problem for TDDFT.61,62 An underestimation of the excited-state dipole moment—combined with
accurate or even overestimated ground-state dipole moments—
leads to underestimated difference dipole moments.62 The
difference dipole moments enter in the expression for the
TPA transition moment in eqn (3) through the term hn|
m| f i
with n = f, which is often the dominating term for noncentrosymmetric molecules.63 Thus, underestimated difference
dipole moments lead to underestimated TPA strengths. For
n a f the element hn|
m| f i in eqn (3) is a transition dipole
moment between two excited states. These elements are likely
to be inaccurate for TDDFT as well since they crucially depend
on the electron density of the excited states, but they are less
studied in the literature. Concluding the discussion of the
TDDFT/CAM-B3LYP TPA strengths, we can say that TDDFT/
CAM-B3LYP generally seems to underestimate TPA strengths
for these molecules with the magnitude varying for different
functionals, molecules and protonation states. There seems to
be a need for benchmarking TDDFT/CAM-B3LYP TPA strengths
of a larger set of medium-sized chromophores—including
different protonation states—against a more accurate method
such as EOM-EE-CCSD in order to obtain a better estimation of
the error inherent to using TDDFT/CAM-B3LYP to calculate the
TPA strength.
Phys. Chem. Chem. Phys., 2015, 17, 19306--19314 | 19311
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PCCP
In this study we have limited the calculations to neutral
chromophores of fluorescent proteins that allow for comparison to available benchmark data. We have not tried to reproduce experimental TPA spectra. To do this in a proper way, one
should not only take into account the form of the lineshape
function for a particular excitation32,36,39 (see Section 2.2) and
the type of experiment that it is compared to (see Section 2.3),
but also temperature effects,48 non-Condon transitions64 and
the environment.37,38 In particular the latter has been shown to
affect both experimental45,46 and calculated37,38 TPA cross
sections in a profound way. Indeed, it has been shown that
different fluorescent proteins with the same chromophore can
have maxima in the TPA spectrum that differ by a factor of 5 in
intensity.45
The discussion of the conversion factor in Section 2 clearly
shows the importance of providing the (correct) details about
the conversion used for determining the TPA cross sections
from TPA strengths in atomic units. The factor of 4 difference
in TPA cross sections between Salem and Brown14 and Nanda
and Krylov13 is probably related to the former using excitation
energies instead of photon energies for o. For the sake of
reproducibility, a good practice is to provide excitation energies
and TPA cross sections in atomic units in addition to macroscopic units, as we have done in Tables 1–4. Not only should the
correct formulae for rotational averaging and conversion to GM
units be given explicitly and applied consistently, one should
also state the type of broadening used and whether the broadening factor G is the full or half width at half maximum. When
comparing to experiments, one needs to ensure that the right
conversion factor is used for the type of experiment employed
(see Fig. 2).31–33,42,43 When comparing to experiments that use
the relative fluorescence method,4,34,46 one could consider
computing the TPA cross section for the reference molecules65
as well and compare the relative cross sections to relative
fluorescence intensities from experiment. This approach corrects for systematic errors in the computational and experimental method in much the same way as for chemical shifts in
NMR spectroscopy, which can be more reliably compared to
experiment than absolute shielding constants.66 In summary,
care should be exercised when comparing calculated and
experimental TPA cross sections, whereas for comparison
between computed results it suffices to use atomic units or to
ensure that the same conversion protocol is used as in the
reference work.
6 Conclusion
We have compared CC2 and TDDFT/CAM-B3LYP TPA strengths
against recent EOM-EE-CCSD benchmark data for a set of
neutral fluorescent protein chromophores. This provides a
much-needed comparison of two methods that allow TPA
calculations on larger (biological) systems (CC2 and TDDFT/
CAM-B3LYP) against a higher-order method (EOM-EE-CCSD). We
have found that CC2 TPA strengths are in good agreement with the
EOM-EE-CCSD cross sections with a small overestimation up to a
19312 | Phys. Chem. Chem. Phys., 2015, 17, 19306--19314
Paper
factor of 1.4 that depends on the molecule. TDDFT/CAM-B3LYP
TPA cross sections, however, are underestimated compared to
coupled-cluster results by a factor of 1.5 to 3 for this set of
molecules, whereas the one-photon oscillator strengths do not
deviate by more than 20%. The underestimation of TDDFT/CAMB3LYP TPA cross sections can be understood in terms of the known
underestimation of excited-state dipole moments of TDDFT
methods.
We have demonstrated how an erroneous assumption on
CC2 TPA cross sections was based on a conversion of the TPA
strengths to macroscopic units that was different from the one
used in the reference work. This stresses the importance of
providing all relevant details in the conversion to macroscopic
units in computational work on TPA, namely (a) the excitation
energy and TPA strength in atomic units, (b) the formulae for both
the conversion and the rotational averaging of the transition
moments, (c) the proper choice of the conversion for comparison
to different types of experiments (see Fig. 2) and (d) the type of
lineshape function used, including the broadening factor and
whether it corresponds to the full or half width at half maximum.
Acknowledgements
This works has received support from the Research Council of
Norway through a Centre of Excellence Grant (Grant No.
179568/V30), from the European Research Council through a
Starting Grant (Grant No. 279619) and from the Norwegian
Supercomputer Program (Grant No. NN4654K). J. K. thanks the
Danish Council for Independent Research (the Sapere Aude
programme), the Danish e-Infrastructure Cooperation (DeIC),
the Lundbeck and the Villum Foundations for financial support.
The authors thank Robert Zaleśny (Wrocław University of Technology, Wroclaw, Poland) for helpful suggestions.
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